Solve for x 3|2x-1|>=21

$3 | 2 x - 1 | \geq 21$

Step 1

Write $3 | 2 x - 1 | \geq 21$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$2 x - 1 \geq 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Divide each term in $2 x \geq 1$ by $2$ and simplify.

Divide each term in $2 x \geq 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{1}{2}$

Divide $x$ by $1$ .

$x \geq \frac{1}{2}$

In the piece where $2 x - 1$ is non-negative, remove the absolute value.

$3 (2 x - 1) \geq 21$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$2 x - 1 < 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 x < 1$

Divide each term in $2 x < 1$ by $2$ and simplify.

Divide each term in $2 x < 1$ by $2$ .

$\frac{2 x}{2} < \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{1}{2}$

Divide $x$ by $1$ .

$x < \frac{1}{2}$

In the piece where $2 x - 1$ is negative, remove the absolute value and multiply by $- 1$ .

$3 (- (2 x - 1)) \geq 21$

Write as a piecewise.

${\begin{cases} 3 (2 x - 1) \geq 21 & x \geq \frac{1}{2} \\ 3 (- (2 x - 1)) \geq 21 & x < \frac{1}{2} \end{cases}$

Simplify $3 (2 x - 1) \geq 21$ .

Apply the distributive property.

${\begin{cases} 3 (2 x) + 3 \cdot - 1 \geq 21 & x \geq \frac{1}{2} \\ 3 (- (2 x - 1)) \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $3$ .

${\begin{cases} 6 x + 3 \cdot - 1 \geq 21 & x \geq \frac{1}{2} \\ 3 (- (2 x - 1)) \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ 3 (- (2 x - 1)) \geq 21 & x < \frac{1}{2} \end{cases}$

Simplify $3 (- (2 x - 1)) \geq 21$ .

Apply the distributive property.

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ 3 (- (2 x) - - 1) \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ 3 (- 2 x - - 1) \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ 3 (- 2 x + 1) \geq 21 & x < \frac{1}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ 3 (- 2 x) + 3 \cdot 1 \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $- 2$ by $3$ .

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ - 6 x + 3 \cdot 1 \geq 21 & x < \frac{1}{2} \end{cases}$

Multiply $3$ by $1$ .

${\begin{cases} 6 x - 3 \geq 21 & x \geq \frac{1}{2} \\ - 6 x + 3 \geq 21 & x < \frac{1}{2} \end{cases}$

Step 2

Solve $6 x - 3 \geq 21$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Add $3$ to both sides of the inequality.

$6 x \geq 21 + 3$

Add $21$ and $3$ .

$6 x \geq 24$

Divide each term in $6 x \geq 24$ by $6$ and simplify.

Divide each term in $6 x \geq 24$ by $6$ .

$\frac{6 x}{6} \geq \frac{24}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{24}{6}$

Divide $x$ by $1$ .

$x \geq \frac{24}{6}$

Simplify the right side.

Divide $24$ by $6$ .

$x \geq 4$

Step 3

Solve $- 6 x + 3 \geq 21$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $3$ from both sides of the inequality.

$- 6 x \geq 21 - 3$

Subtract $3$ from $21$ .

$- 6 x \geq 18$

Divide each term in $- 6 x \geq 18$ by $- 6$ and simplify.

Divide each term in $- 6 x \geq 18$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \leq \frac{18}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \leq \frac{18}{- 6}$

Divide $x$ by $1$ .

$x \leq \frac{18}{- 6}$

Simplify the right side.

Divide $18$ by $- 6$ .

$x \leq - 3$

Step 4

Find the union of the solutions.

$x \leq - 3$ or $x \geq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 3 or x \geq 4$

Interval Notation:

$(- \infty, - 3] \cup [4, \infty)$

Step 6

Solve for x 3|2x-1|>=21

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